Untersuchungen über die Grundlagen der Mengenlehre I.¶
Zermelo, E. (1908). Untersuchungen über die Grundlagen der Mengenlehre I. Mathematische Annalen, 65(2), 261-281.
Cited by¶
6 citations across 6 artifacts.
Each citation links to the sentence it supports in the citing article.
Primes¶
- Infinity
- Subsequent foundational work — Zermelo's (1908) axiomatization
This sourceFoundational axiomatization of set theory; axiom of choice and well-ordering principle.
- Subsequent foundational work — Zermelo's (1908) axiomatization
- Paradox
- T3: Set-Theoretic Foundations and Axiom Choice
This sourceFoundational axiomatization of set theory; axiom of choice and well-ordering principle.
- T3: Set-Theoretic Foundations and Axiom Choice
- Set and Membership
- In classical (ZFC) set theory
This sourceFoundational axiomatization of set theory; axiom of choice and well-ordering principle.
- In classical (ZFC) set theory
- Tolerance Paradox
- The resolution is precisely a meta-level exception: Zermelo–Fraenkel replaces unrestricted comprehension with restricted (separation) comprehension, which only forms {x ∈ A : P(x)} — subsets of an already-existing set A.
This sourceIntroduces the axiom of separation (restricted comprehension), blocking Russell's paradox while preserving ordinary set construction.
- The resolution is precisely a meta-level exception: Zermelo–Fraenkel replaces unrestricted comprehension with restricted (separation) comprehension, which only forms {x ∈ A : P(x)} — subsets of an already-existing set A.
- Well-Foundedness (Well-Ordering)
- … ring is Noetherian) is a core algebraic-geometry foundational result that depends on this; Zorn's lemma (equivalent to AC) supplies well-founded-style maximal elements in inductively-ordered sets; regular cardinals and stationary sets in set theory rely on well-foundedness arguments for their existence and uniqueness.
This sourceAxiomatisation of set theory; precursor of the foundation axiom.
- … ring is Noetherian) is a core algebraic-geometry foundational result that depends on this; Zorn's lemma (equivalent to AC) supplies well-founded-style maximal elements in inductively-ordered sets; regular cardinals and stationary sets in set theory rely on well-foundedness arguments for their existence and uniqueness.
Domain-specific¶
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